080223092400-PRSA_CODES from Continuation of the Stieltjes series to the large regime by finite-part integration

We devise a prescription to use a novel convergent expansion in the strong-asymptotic regime for the Stieltjes integral and its generalizations (Galapon EA. 2017 <i>Proc. R. Soc A</i> <b>473</b>, 20160567) to sum the associated divergent series of Stieltjes across all asymptotic regimes. The novel expansion makes use of the divergent negative-power moments, which we treated as Hadamard’s finite part integrals. The result allowed us to compute the ground state energy of the quartic, sextic anharmonic oscillators as well as the $\mathcal {PT}$ symmetric cubic oscillator, and the funnel potential across all perturbation regimes from a single expansion that is built from the divergent weak-coupling perturbation series and incorporates the known leading-order strong-coupling behaviour of the spectra.